Let f(x) be a function satisfying f(x + y) = f(x) f(y) for all x, y ∈ N such that f(1) = 2 :
What is \(\displaystyle\sum_{x=1}^{6} 2^{x}\,f(x)\) equal to?
5460
Step 1 – Identify f(x):
The relation \(f(x+y)=f(x)f(y)\) with \(f(1)=2\) gives \(f(x)=2^{x}\).
Step 2 – Substitute and simplify:
\(\displaystyle\sum_{x=1}^{6} 2^{x}\,f(x)=\sum_{x=1}^{6} 2^{x}\cdot 2^{x}=\sum_{x=1}^{6} 4^{x}\)
Step 3 – Sum the geometric series:
\(\displaystyle\sum_{x=1}^{6} 4^{x}=\dfrac{4(4^{6}-1)}{4-1}=\dfrac{4\times 4095}{3}=\dfrac{16380}{3}=5460\)
Hence the value is 5460.
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